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Episode 2 Supports

  • Episode Description

    Exploring: Keoni and Sasha use points that share an x-value to explain why increasing the p-value in the equation y = x2/(4p) results in the parabola getting wider.

     

  • Students’ Conceptual Challenges

    When asked to justify why the parabolas get wider as the p-value increases, Sasha restates her claim. She and Keoni struggle to identify evidence that can support their claim [2:11-2:19].

     

    • The teacher asks them to attend to the y-values of the labeled points. Keoni states that the y-values decrease as the p-value increases. When prompted, Sasha indicates how the decreasing y-values impact the relative widths of the parabolas. Keoni and Sasha continue to build on the precision of their language in this episode and the next.

  • Focus Questions

    For use in a classroom, pause the video and ask these questions:

     

    1. [Pause video at 0:53]. How can we confirm that the three labeled points are on each parabola?

     

    2. [Pause video at 3:02]. What do you think that Keoni means when he says “that it is getting wider by staying…”?

  • Supporting Dialogue

    Invite students to engage in stating and justifying mathematical claims with another student:

     

    • Work with a partner to use the evidence in this episode in order to make a claim about the how changing the p-value changes the width of the parabola.

    • Work with a partner to build justification for your claim from the evidence in the episode.

  • Math Extensions

    1. Find the coordinates of points on each of the three parabolas when the x-value is 1.

     

    2. Considering the ordered pairs that you found, what do you notice about the y-values when the p-value increases? How does that impact the shape of the parabola?

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Mathematics in this Lesson

Lesson Description

Targeted Understanding

CC Math Standards

CC Math Practices

Lesson Description

 

Sasha and Keoni use algebraic and geometric thinking to form three arguments that justify why a parabola gets wider on the coordinate grid as the p-value in y = x2/(4p) increases.

Targeted Understandings

 

This lesson can help students:

 

  • Understand why, when the p-value in y =  x2/(4p) increases the parabolas appear to be getting wider on the same coordinate grid, by advancing three different arguments:

  • When x is a fixed value (e.g., when x = 2 and y = 1/p, then as p increases, y decreases, which means the parabola will get wider.
  • When y is a fixed value (e.g., when y = 4 and 4 = x2/(4p) or 4√p = x), then as p increases, the square root of p increases. This means that 4√p, which is x, also increases. Thus, as p increases, so does x, which means the parabola will get wider.
  • A “special point” for a parabola (i.e., a point on the parabola that is aligned horizontally with the focus) can be expressed in general as (2p, p). Thus, as p increases, the special point for the parabola will be growing faster outward than it grows in the upward direction. This means the parabolas will appear to get wider on the same coordinate grid.

Common Core Math Standards

• CCSS.M.HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Sasha and Keoni first notice key features of the graphs of three quadratic functions (all with a vertex at the origin but with different p-values). For example, they notice that when the x-value is the same for all three functions (e.g., when x = 2), then the y-values of the points decrease as the distance between the vertex and focus (the p-value) increases. They observe visually on the graph how decreasing the y-value but keeping the x-value constant forces the parabola to become wider. They then link this observation to the algebraic expression for the parabolas. By fixing the x-value as 2, the equation y = x2/(4p) becomes y = 4/(4p), which is y = 1/p. Sasha and Keoni can see algebraically that as p increases, y decreases. Comparing the algebraic and graphical representations of these quadratic functions provides a way to help understand why increasing the p-value in y =  x2/(4p)  means the parabolas will appear to be getting wider on the same coordinate grid.

Common Core Math Practices

 

CCSS.Math.Practice.MP7. Look for and make use of structure.

In this lesson, Sasha and Keoni construct three different viable arguments to explain why  increasing the p-value in y = x2/(4p) means the parabolas will appear to be getting wider on the same coordinate grid. Each argument relies on a consideration of different features of the graphs: (a) fixing the x-value in Episodes 2-3; (b) fixing the y-value in Episode 4; and (c) considering special points in Episodes 5-6. Noticing a relationship is much easier than creating an argument to explain why the relationship holds! To accomplish the latter, Sasha and Keoni express their ideas to each other [Episode 2, 2:33 – 3:42], even when it’s difficult to do so. They also justify their conclusions [Episode 2, 5:02 – 5:33] and respond to each others’ explanations [Episode 6, 1:26 – 1:56].